On sums of Ramanujan sums

TL;DR

This paper derives asymptotic formulas for sums involving powers of Ramanujan sums over large ranges, advancing understanding of their distribution and behavior in number theory.

Contribution

It provides new asymptotic formulas for sums of powers of Ramanujan sums, extending previous results in analytic number theory.

Findings

Asymptotic formulas for sums of Ramanujan sums are established.

Results apply to large ranges with specific growth conditions.

The work enhances understanding of Ramanujan sum distributions.

Abstract

Let $c_q(n)$ denote the Ramanujan sum modulo $q$, and let $x$ and $y$ be large reals, with $x = o(y)$. We obtain asymptotic formulas for the sums $$\sum_{n \le y}(\sum_{q \le x} c_q(n))^k \qquad (k = 1, 2).$$